Hypercontractivity on High Dimensional Expanders: a Local-to-Global Approach for Higher Moments

TL;DR

This paper develops a new hypercontractivity theory for high dimensional expanders, enabling advanced analysis of Boolean functions and revealing structural properties that extend classical results to complex, non-product settings.

Contribution

It introduces a novel local-to-global method and an explicit Fourier basis for HDX, advancing the understanding of higher moments and Boolean functions on these complexes.

Findings

A tight analog of the KKL Theorem for HDX

Characterization of non-expanding sets in HDX

New tools for analyzing higher moments in complex structures

Abstract

Hypercontractivity is one of the most powerful tools in Boolean function analysis. Originally studied over the discrete hypercube, recent years have seen increasing interest in extensions to settings like the $p$-biased cube, slice, or Grassmannian, where variants of hypercontractivity have found a number of breakthrough applications including the resolution of Khot's 2-2 Games Conjecture (Khot, Minzer, Safra FOCS 2018). In this work, we develop a new theory of hypercontractivity on high dimensional expanders (HDX), an important class of expanding complexes that has recently seen similarly impressive applications in both coding theory and approximate sampling. Our results lead to a new understanding of the structure of Boolean functions on HDX, including a tight analog of the KKL Theorem and a new characterization of non-expanding sets. Unlike previous settings satisfying hypercontractivity, HDX can be asymmetric, sparse, and very far from products, which makes the application of traditional proof techniques challenging. We handle these barriers with the introduction of two new tools of independent interest: a new explicit combinatorial Fourier basis for HDX that behaves well under restriction, and a new local-to-global method for analyzing higher moments. Interestingly, unlike analogous second moment methods that apply equally across all types of expanding complexes, our tools rely inherently on simplicial structure. This suggests a new distinction among high dimensional expanders based upon their behavior beyond the second moment.